First slide

Methods of integration

Question

$\int \frac{1}{x^{1 / 5} {(1 + x^{4 / 5})}^{1 / 2}} d x$ , is

Moderate

A

$\sqrt{1 + x^{4} / 5} + C$
B

$\frac{5}{2} \sqrt{1 + x^{4 / 5}} + C$
C

$x^{4 / 5} \sqrt{1 + x^{4 / 5}} + C$
D

none of these

Solution

Let $I = \int \frac{1}{x^{1 / 5}} \frac{1}{{(1 + x^{4 / 5})}^{1 / 2}} d x$
$\begin{array}{l} \Rightarrow I = \frac{5}{4} \int \frac{1}{{(1 + x^{4 / 5})}^{1 / 2}} (\frac{4}{5} x^{- 1 / 5}) d x \\ \Rightarrow I = \frac{5}{4} \int \frac{1}{{(1 + x^{4 / 5})}^{1 / 2}} d (1 + x^{4 / 5}) = \frac{5}{2} {(1 + x^{4 / 5})}^{1 / 2} + C \end{array}$

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